Making a smooth transition from full alpha opacity to full transparency
Your issue is that you are applying the gradient to the color of the transparent shader, instead of using it to control how the two shaders are mixed. Since your factor slot is empty, they are just being mixed with a factor of 0.5 across the whole surface.
My result:

The nodes:

Explanation
To transition from a color to transparent, you essentially want to mix the two shaders, here a transparent BSDF and an emission shader, with the mix factor increasing along the gradient. For a simple gradient along the X axis, I used a color ramp, but you could apply the same process with any other gradient texture. Here, I want a value from 0 to 1 along the color ramp based on the X position on the object, so I plug in the X component of the Object coordinates as the factor to the color ramp, and use that to control the mixing.